H

electrons. This pairing leaves two lone pairs of electrons in the outer-shell, the orbitals of which point to the vertices of an approximately regular tetrahedron formed by the lone pairs and the OH bonds. The resulting tetra-hedral structure has two positively charged sites at one side and two negatively charged sites at the other. It will readily attach itself by hydrogen bonds to four neighbouring molecules, two at the negatively charged sites and two at the positively charged sites. In its usual form, ice demonstrates an almost perfect tetra-hedral arrangement of bonds with a distance of about 0.276 nm between neighbouring oxygen atoms. There is much unfilled space in the crystal, which accounts for the low density of ice.

When ice melts, a high degree of hydrogen bonding persists in the resulting liquid. In spite of extensive investigation by a variety of techniques such as X-ray diffraction, thermo-chemical determination, and infrared and Raman spectroscopy, the structural nature of liquid water is still to be completely resolved. There are, broadly speaking, two distinct types of model: those which involve distortion but not breaking of hydrogen bonds, and a second type in which unbonded detached water molecules exist in addition to the hydrogen-bonded structures. Of the former type, the model which is considered to be the most acceptable is one in which all the water molecules continue to be hydrogen-bonded to their four neighbours, but the intermolecular links are bent or stretched to give an irregular framework. Such distorted networks are known to exist in some of the denser forms of ice.

Many proposed structures for water involve mixtures of structured material and free water molecules. One of the most highly developed theories encompasses the so-called 'flickering cluster' concept of water structure. The model is based on the cooperative nature of hydrogen bonding. The formation of one hydrogen bond on a water molecule leaves the molecule more susceptible to further hydrogen bonding, and similarly when one bond breaks there is a tendency for large groups of bonds to break. As a result, clusters of ice-like hydrogen-bonded material are imagined to be suspended in a fluid of unbonded water (Fig. 6.23). Because of the continual formation and rupture of hydrogen bonds throughout the liquid, these clusters have only a temporary existence, and are aptly described by the term 'flickering'.

Most of the models proposed for the structure of water, only two of which have been considered here, can account for some, but not all of the physical and thermodynamic anomalies which have been observed with water.

The flickering cluster model can be used to describe possible structural changes that occur when nonpolar and polar solutes are dissolved in water. A nonpolar molecule or portion of a molecule tends to seek out the more ice-like regions within the water. Such regions, as we have seen, contain open structures into which the nonpolar molecules may fit without breaking hydrogen bonds or otherwise disturbing the surrounding ice-like material. In solution, therefore, hydrophobic molecules tend always to be surrounded by structured water. This concept is important in discussing interactions between nonpolar molecules in aqueous solution, such as those that occur in micelle formation. The interaction of hydrocarbons in aqueous solution was first thought to arise simply as consequence of the van der Waals forces between the hydrocarbon molecules. It was later realised, however, that changes in the water structure around the nonpolar groups must play an important role

Figure 6.23 Water clusters with unassociated water molecules around them.

in the formation of bonds between the nonpolar molecules - the so-called hydrophobic bonds. In fact the contribution from the van der Waals forces is only about 45% of the total free energy of formation of a hydrophobic bond. When the nonpolar groups approach each other until they are in contact, there will be a decrease in the total number of water molecules in contact with the nonpolar groups. The formation of the hydrophobic bond in this way is thus equivalent to the partial removal of hydrocarbon from an aqueous environment and a consequent loss of the ice-like structuring which always surrounds the hydrophobic molecules. The increase in entropy and decrease in free energy which accompany the loss of structuring make the formation of the hydrophobic bond an energetically favourable process.

There is much experimental evidence to support this explanation for the decrease in À G. Thus the enthalpy of micelle formation becomes more negative as the temperature is increased; a fact which was attributed to a reduction in water structure as temperature is increased. Nuclear magnetic resonance (NMR) measurements indicate an increase in the mobility of water protons at the onset of micellisation. The addition of urea, a water-structure-breaking compound, to surfactant solutions leads to an increase of cmc, again indicating the role of water structure in the micellisation process. An alternative explanation of the free energy decrease emphasises the increase in internal freedom of the hydrocarbon chains which occurs when these chains are transferred from the aqueous environment, where their motion is restrained by the hydrogen-bonded water molecules, to the interior of the micelle. It has been suggested that the increased mobility of the hydrocarbon chains, and of course their mutual attraction, constitute the principal hydro-phobic factor in micellisation.

6.3.2 Theories of micelle formation

Two general approaches have been employed in attempting to describe the process of micellisation. In one of these, the phase separation model, the cmc is assumed to represent the saturation concentration of the unassoci-ated molecules and the micelles are regarded as a distinct phase which separates out at the cmc. In the alternative approach, the micelle and associated monomers are assumed to be in an association-dissociation equilibrium to which the law of mass action may be applied. Neither of these models is rigorously correct, although the mass action approach seems to give a more realistic description of micellisa-tion, and thus will be considered in more detail.

The aggregation process may in its simplest form be described by

Equation (6.20) represents the formation of a cationic micelle Mp+ from N surfactant ions D + and (N - p) firmly held counterions X-. Whenever the thermodynamics of a process is under consideration, it is important to define the standard states of the species. In this example, the standard states are such that the mole fractions of the ionic species are unity and the solution properties are those of the infinitely dilute solutions. The equilibrium constant Km may be written in the usual way

where activity coefficients have been neglected. The analogous equation for nonionic micelles is of a simpler form since counterion terms and charges need not be considered.

effect (although it can be detected experimentally from surface-tension measurements) and for most purposes it is reasonable to assume that the monomer concentration remains constant at the cmc value. A second point of interest illustrated by the mass action treatment concerns the predicted sharpness of the cmc. It is readily shown by calculations that combinations of low values of N and Km lead to gradual changes of slope of the cmc region, while larger values for both of these parameters give sharp inflections. The cmc, rather than being an exact concentration, is often a region of concentration over which the solution properties exhibit a gradual change and hence is often difficult to locate exactly.

6.3.3 Micellar structure Critical packing parameter

The shape of the micelle formed by a particular surfactant is influenced to a large extent by the geometry of the surfactant molecule, as can be seen if we consider the packing of space-filling models of the surfactants. The dimensionless parameter of use in these considerations is called the critical packing parameter (CPP) and is defined as v

Equation (6.21) and (6.22) are important in that they can be used to predict the variation of both monomers and micelles with total solution concentration.

Figure 6.24 shows the result of such a calculation for a model system. It illustrates several important points about the micellisation process. According to the mass action treatment, the monomer concentration decreases very slightly above the cmc: this is a very small